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ENERGY                                       1;
Denoting the kinetic energy by T and putting the defii] tion into analytical language we obtain
r-- rt-m'-
l r^dv
Jo  dt
Therefore the kinetic energy of a particle equals one-half the prc net of the mass by the square of its velocity. Since both m ai v2 are positive, kinetic energy must be a positive magnituc The kinetic energy of a system of particles, therefore, eqm the arithmetic sums of the kinetic energies of the individu particles. Thus
T =
When all the particles of the system have the same veloci
where M is the total mass of the system.
154. Work Done in Increasing the Velocity of a Particle. — the velocity of a particle is increased from VQ to v then t work done against the kinetic reaction equals the incre£ in the kinetic energy of the particle. This will be seen frc the following analysis :
™ — ml
J Vn
at vdv
' »0
= 2" WVD   — 2 "WQ    I                                            /T-
= T-T0.        j                     ^
* The first negative sign indicates the fact that T is the work done agai and not by the kinetic reactions.
The second negative sign belongs to the kinetic reaction as it was explai in Chapter VI.